What are the Sets in Mathematics?
Sets are defined as the collection of well-defined data. In Math, a Set is a tool that helps to classify and collect data belonging to the same category, even though the elements used in sets are all different from each other, they all are similar as they belong to one group. For instance, a set of different outdoor games, say set A= {Football, basketball, volleyball, cricket, badminton} all the games mentioned are different, but they all are similar in one way as they belong to the same group (outdoor games).
The set is denoted as a capital letter, for example, set A, set B, etc., and the elements belonging to the set are denoted as a small letter, and they are kept in curly brackets {}, for example, set A= {a, b, c, d}, as it is clear that a, b, c, d belong to set A, it can be written a ∈ A, do p belong to set A? No. Therefore, it will be written as, p∉ A.
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Representation of Sets
Sets can be represented in two ways, one is known as the Roster form and the other is famous as the Set-Builder form, these two forms can be used to represent the same data, but the style varies in both cases.
Roster Form
In Roster Form, the elements are inside {}⇢ Curly brackets. All the elements are mentioned inside and are separated by commas. Roster form is the easiest way to represent the data in groups. For example, the set for the table of 5 will be, A= {5, 10, 15, 20, 25, 30, 35…..}.
Properties of Roster Formrelations of Sets:
- The arrangement in the Roster form does not necessarily to be in the same order every time. For example, A= {a, b, c, d, e} is equal to A= {e, d, a, c, b}.
- The elements are not repeated in the set in Roster form, for example, the word “apple” will be written as, A= {a, p, l, e}
- The Finite sets are represented either with all the elements or if the elements are too much, they are represented as dots in the middle. The infinite sets are represented with dots in the end.
Set-Builder Form
In Set-builder form, elements are shown or represented in statements expressing relations among elements. The standard form for Set-builder, A= {a: statement}. For example, A = {x: x = a3, a ∈ N, a < 9}
Properties of Set-builder form:
- In order to write the set in Set- builder form, the data should follow a certain pattern.
- Colons (:) are necessary in Set-builder form.
- After colon, the statement is to be written.
Order of the Set
The order of the Set is determined by the number of elements present in the Set. For example, if there are 10 elements in the set, the order of the set becomes 10. For finite sets, the order of the set is finite, and for infinite sets, the order of the set is infinite.
Sample Problems
Question 1: Determine which of the following are considered assetsin and which are not.
- All even numbers on the number line.
- All the good basketball players from class 9th.
- The bad performers from the batch of dancers.
- All prime numbers from 1 to 100.
- Numbers that are greater than 5 and less than 15.
Answer:
Sets are not those bunches or groups where some quality or characteristic comes in the picture. Therefore,
- “All even numbers on the number line” is a set.
- “All the good basketball players from class 9th” is not a Set as “good” is a quality which is involved.
- “The bad performers from the batch of dancers” cannot be a Set since “bad” is a characteristic.
- “All prime numbers from 1 to 100” is a Set.
- “Numbers that are greater than 5 and less than 15” is a Set.
Question 2: Represent the following information inSet-Builder the Roster form.
- All Natural numbers.
- Numbers greater than 6 and less than 3.
- All even numbers from 10 to 25.
Answer:
The Roster form for the above information,
- Set A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11……}
- Set B = {} ⇢ Null set, since there are no numbers greater than 6 and less than 3.
- Set C = {10, 12, 14, 16, 18, 20, 22, 24}
Question 3: Express the given information in the Set-Builder form.
- Numbers that are greater than 10 and less than 20.
- All Natural numbers greater than 25.
- Vowels in English Alphabet.
Answer:
The Set-Builder form for the above information,
- A = {a: a∈ N and 10 < a < 20}
- B = {b: b∈ N and b > 25}
- C = {c: c is the vowel of English Alphabet}
Question 4: Convert the following Sets given in Roster form into Set-Builder form.
- A = {b, c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}
- B = {2, 4, 6, 8, 10}
- C = {5, 7, 9, 11,13, 15, 17, 19}
Answer:
The Set- builder form for the above Sets,
- A = {a: a is a consonant of the English Alphabet}
- B = {b: b is an Even number and 2 ≤ b ≤10}
- C = {c: c is an odd number and 5 ≤ c ≤ 19}
Question 5: Give an example of the following types of Sets in both Roster form and Set-builder form.
- Singular Set.
- Finite Set.
- Infinite Set.
Solution:
The Examples can be taken as per choice since there can be a infinite number of examples for any of the above Sets,
- Singular Set
Roster Form: A = {2}
Set- builder form: A= {a: a∈N and 1<a<3}
- Finite Set
Roster Form: B = {0,1, 2, 3, 4, 5}
Set-builder form: B = {b: b is a whole number and b<6}
- Infinite Set
Roster Form: C = {2, 4, 6, 8, 10, 12, 14, 16…..}
Set- builder form: C= {c: c is a Natural and Even number}
Question 6: What is the order of the given sets,
- A = {7, 14, 21, 28, 35}
- B = {a, b, c, d, e, f, g….x, y, z}
- C = {2, 4, 6, 8, 10, 12, 14……}
Answer:
The order of the set tells the number of element present in the Set.
- The order of Set A is 5 as it has 5 elements.
- The order of set B is 26 as the English Alphabet have 26 letters.
- The order of set C is infinite as the set has the infinite number of elements.
Question 7: Express the given Sets in Roster form,
- A = {a: a = n/2, n ∈ N, n < 10}
- B = {b: b = n2, n ∈ N, n ≤ 5}
Answer:
Representing the above Set-builder sets in Roster form,
- A = {1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2}
- B = {1, 4, 9, 16, 25}
Conclusion
The representation of a set is a fundamental concept in mathematics that allows us to the describe and manipulate collections of the distinct objects or elements. The Sets can be represented in the various forms including the roster form, set-builder form and Venn diagrams. Understanding these different representations helps in the visualizing and solving problems related to the unions, intersections, subsets and other set operations. Mastery of set representation is essential for the students and professionals working in the fields like mathematics, computer science and logic where sets form the basis for the more complex concepts.
FAQs on Representation of a Set
What is the difference between roster form and set-builder form?
The Roster form lists all the elements of the set explicitly while set-builder form describes the properties that characterize the elements of the set.
Can a set have duplicate elements?
No, a set cannot have duplicate elements. By definition all elements in the set are unique.
What does it mean for a set to be finite or infinite?
A finite set has a specific number of the elements whereas an infinite set has an unbounded number of the elements.
How is an empty set represented?
An empty set, which contains no elements is represented by the symbol ∅ or by a pair of braces with the nothing inside {}.